# How to choose between two uncertainty calculation method?

Hello,
I have made an experimental work and I am ask to provide a formal experimental error analysis. I have a difficulty to choose the appropriate analysis way and would like to have some advices or explanations.
I have measured a temperatures at the intlet (Ti) and outlet (To) of a heat exchanger and the mass flow rate of the fluid. I can calculate the power:
Q = mC (To-Ti)
From what I learn on uncertainty calculations, I am a little bit confused. I finally found that I could calculate my uncertainty in two ways:
(1): $\frac{ΔQ}{Q}$ = $\frac{Δm}{m}$+ $\frac{2ΔT}{To-Ti}$

(2): $\frac{ΔQ}{Q}$ =$\sqrt{\left(\frac{Δm}{m}\right)^2+\left(\frac{2ΔT}{To-Ti}\right)^2}$

Δm: accuracy of the mass flowrate meter (documentation of the supplier)
ΔT: accuracy of the mass flowrate meter (value given by the calibration center)
C is admit as a constant

I made just one measurement. Which one of the above equations must I use and why?

Actually, I derived heat transfer coefficients (built a curve) from several measurements, by changing the flowrate m and the inlet temperature Ti. But no test has been repeated in the same conditions. Is the first method or the second one am I going to use for the "error analysis"?
Thank you in advance.
P.S. Please, even if you do not have a time, even a short answer would be useful for me.

## Answers and Replies

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Hello,
I have made an experimental work and I am ask to provide a formal experimental error analysis. I have a difficulty to choose the appropriate analysis way and would like to have some advices or explanations.
I have measured a temperatures at the intlet (Ti) and outlet (To) of a heat exchanger and the mass flow rate of the fluid. I can calculate the power:
Q = mC (To-Ti)
From what I learn on uncertainty calculations, I am a little bit confused. I finally found that I could calculate my uncertainty in two ways:
(1): $\frac{ΔQ}{Q}$ = $\frac{Δm}{m}$+ $\frac{2ΔT}{To-Ti}$

(2): $\frac{ΔQ}{Q}$ =$\sqrt{\left(\frac{Δm}{m}\right)^2+\left(\frac{2ΔT}{To-Ti}\right)^2}$

Δm: accuracy of the mass flowrate meter (documentation of the supplier)
ΔT: accuracy of the mass flowrate meter (value given by the calibration center)
C is admit as a constant

I made just one measurement. Which one of the above equations must I use and why?

Actually, I derived heat transfer coefficients (built a curve) from several measurements, by changing the flowrate m and the inlet temperature Ti. But no test has been repeated in the same conditions. Is the first method or the second one am I going to use for the "error analysis"?
Thank you in advance.
P.S. Please, even if you do not have a time, even a short answer would be useful for me.

If the two accuracies are independent and the numbers you have giving us are standard deviations then use equation 2. The reason is that independent errors tend to cancel to some degree, so to "add" them correctly one must use the formula you show there.

Thank you ImaLooser for your explanation. I assume that the accuracies are independant. But I don't know if the accuracy of the intruments are standard deviation (I always think that they are the maximal error; I may be wrong). Actually, for a long time, I do not realise that the accuracy of an instrument can be given as a standard deviation. So, in case, it is the maximal error, should I keep equation (2)?
Thank you.

Thank you ImaLooser for your explanation. I assume that the accuracies are independant. But I don't know if the accuracy of the intruments are standard deviation (I always think that they are the maximal error; I may be wrong). Actually, for a long time, I do not realise that the accuracy of an instrument can be given as a standard deviation. So, in case, it is the maximal error, should I keep equation (2)?
Thank you.
If it is standard deviation then use 2. If it is systematic error, then use 1.

Systematic error is repeatable error. The system always gives exactly the same answer, but the answer is inaccurate in the same way every time.

If it is some unknown kind of error, then use 1. It will always be greater than 2, so you are playing it safe.

Thank you very much. I understand now the difference.